Download - Digital Filter Structure
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Digital Filter Structure
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Block Diagram Representation
Basic Building Blocks
Analysis of Block Diagrams
Equivalent Structures Basic FIR Digital Filter Structures
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Digital Filter Structure
The convolution sum description of an LTI discrete-time system
can, in principle, be used to implement the system.
For example, an FIR system can be implemented using the
convolution sum which is a finite sum of products:
[ ]=0
For an IIR finite-dimensional system this approach is not practical as
the impulse response is of infinite length. However, a direct
implementation of the IIR finite-dimensional system is practical.
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The actual implementation of an LTI digital filter can be either in
software or hardware form, depending on applications.
In either case, the signal variables and the filter coefficientscannot be
represented with infinite precision.
Thus, a direct implementation of a digital filter based on eitherthe
difference equation or the finite convolution sum may not provide
satisfactory performance due to the finite precision arithmetic.
It is thus of practical interest to develop alternate realizations and
choose the structure that provides satisfactory performance under
finite precision arithmetic.
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Block Diagram Representation
A structural representation using interconnected basic building
blocksis the first step in the hardware or software implementation
of an LTI digital filter.
The structural representation provides the key relations between
some pertinent internal variables with the input and output that inturn provides the key to the implementation.
In time-domain, the input-output relations of an LTI digital filter is
given by the convolution sum [ ]=0
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For the implementation of an LTI digital filter, the input-output
relationship must be described by a valid computational algorithm.
To illustrate what we mean by a computational algorithm, consider the
causal first-order LTI digital filter shown below.
The filter is described by the difference equation
Y[n]=-d1y[n-1]+p0x[n]+p1x[n-1]
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The computational algorithm of an LTI digital filter
can be conveniently represented in block diagramform using the basic building blocks shown below:
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Advantages of Block Diagram
Representation
(1) Easy to write down the computational algorithm byinspection
(2) Easy to analyze the block diagram to determine the
explicit relation between the output and input(3) Easy to manipulate a block diagram to derive other
equivalent block diagrams yielding different
computational algorithms
(4) Easy to determine the hardware requirements
(5) Easy to develop block diagram representations from
the transfer function directly
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Analysis of Block Diagrams
Question: How to analyze the block diagrams?
Analysis is carried out by
Writing down the expressions for the output signals of
each adder as a sum of its input signalsDeveloping a set of equations relating the filter input and
output signals in terms of all internal signals
Eliminating the unwanted internal variables then results in
the expression for the output signal as a function of theinput signal and the filter parameters that are the
multiplier coefficients.
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Consider the single-loop feedback structure shown
below
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Direct Form FIR Digital Filter Structures
Consider an FIR filter with N=4.
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The direct form structure shown on the previous slide
is also known as a transversal filter.
Transversal filter: a non-recursive filter using atapped delay line to implement the basic filter
equation.
The transpose of the direct form structure is indicated
below.